SamirTafesh:
the best way to program it is by using the Arduino IDE (an inteface that you are familiar with), and you can write your code as if you are writing for Arduino, all you need is to add the BOARD library in the IDE interface
If you look at the tutorial that is already what they're trying to do, the firmware, ESPEasy, is an Arduino sketch they're trying to upload using the esp8266 core.
INTP:
I'm not seeing what tutorials are saying I should be seeing as far as number format.
Please explain which part of the tutorial you're talking about. I don't see anything about the AT firmware version on that tutorial you linked. Those are instructions for uploading their own custom firmware which will overwrite the AT firmware currently on your module, thus your current firmware version is irrelevant to that tutorial.
INTP:
What firmware version do I have and what's the most current one?
AT+GMR
AT version:0.60.0.0(Jan 29 2016 15:10:17)
SDK version:1.5.2(7eee54f4)
Ai-Thinker Technology Co. Ltd.
AI-Thinker is the manufacturer of the module, not the ESP8266 chip, apparently they have their own AT firmware version that allows control of the GPIO pins via AT commands, something the Espressif firmware doesn't do. I've been unable to find much information about their firmware because the AI-Thinker website is all in Chinese and refuses to Google translate.
The more recent versions of the AT firmware from Espressif, the manufacturer of the ESP8266 chip, require more than the 512kB flash usually found on the ESP-01 modules. I previously assumed that all ESP-01 had 512kB flash but I've been informed some have 1MB.
I think the latest Espressif AT firmware that supports 512kB flash is 0.40 but I have no clue about the AI-Thinker firmware. It's possible you already have the latest that your module will support.
INTP:
how is the ESP8266 physically connected? Through Uno TX/RX as I already have it? Through an FTDI (I have an FTDI board using FT232 but it's 5V)?
It's easiest to use an FTDI board or similar. Some people have used their Arduinos as a FTDI board to program these things by bypassing the microcontroller. Some FTDI breakout boards have a jumper on them to change from 5V to 3.3V.
INTP:
And how do I figure out all of the parameters after board select? I have no idea what kind of flash mem this module has and what SPIFFs to pick, etc.
That is extremely confusing. There are so many versions of these modules, and very few of the sellers provide any useful specs. Even knowing it's an "ESP-01" may not be enough because that appears to just refer to the general form factor and the flash chip may vary. I struggled with this for a long time when I was trying to use the Espressif Flash Download Tool to update my AT firmware. You need to fill in all that information on the flash. I am using the modules with the shield so I couldn't see the part number on the flash chip. Finally I discovered that program can auto-detect the flash chip configuration, something not mentioned in any of the tutorials I was using. The esp8266 Arduino core has a similar feature at File > Examples > ESP8266 > CheckFlashConfig. You will of course need to upload it to your module to use it so that's a bit of a catch 22. I don't know what the consequences of doing this with the wrong settings are but at least the modules are dirt cheap if you somehow did mess it up.